Improving dimension estimates for Furstenberg-type sets
In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α....
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Artículo publishedVersion |
| Publicado: |
2010
|
| Materias: | |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v223_n2_p672_Molter_oai |
| Aporte de: |
| id |
I28-R145-paper_00018708_v223_n2_p672_Molter_oai |
|---|---|
| record_format |
dspace |
| spelling |
I28-R145-paper_00018708_v223_n2_p672_Molter_oai2024-08-16 Molter, U. Rela, E. 2010 In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. application/pdf http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar Adv. Math. 2010;223(2):672-688 Dimension function Furstenberg sets Hausdorff dimension Improving dimension estimates for Furstenberg-type sets info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v223_n2_p672_Molter_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (UBA) |
| topic |
Dimension function Furstenberg sets Hausdorff dimension |
| spellingShingle |
Dimension function Furstenberg sets Hausdorff dimension Molter, U. Rela, E. Improving dimension estimates for Furstenberg-type sets |
| topic_facet |
Dimension function Furstenberg sets Hausdorff dimension |
| description |
In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved. |
| format |
Artículo Artículo publishedVersion |
| author |
Molter, U. Rela, E. |
| author_facet |
Molter, U. Rela, E. |
| author_sort |
Molter, U. |
| title |
Improving dimension estimates for Furstenberg-type sets |
| title_short |
Improving dimension estimates for Furstenberg-type sets |
| title_full |
Improving dimension estimates for Furstenberg-type sets |
| title_fullStr |
Improving dimension estimates for Furstenberg-type sets |
| title_full_unstemmed |
Improving dimension estimates for Furstenberg-type sets |
| title_sort |
improving dimension estimates for furstenberg-type sets |
| publishDate |
2010 |
| url |
http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v223_n2_p672_Molter_oai |
| work_keys_str_mv |
AT molteru improvingdimensionestimatesforfurstenbergtypesets AT relae improvingdimensionestimatesforfurstenbergtypesets |
| _version_ |
1809357037278068736 |